Uniform Ergodicity Research Articles

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l 1 . Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Schwach irreduzible Markoff-Operatoren

For any weakly irreducible Markov operatorT on the space of continuous functions on a compact space and any continuous functionf we show that pointwise convergence of (Tnf) already implies uniform convergence. This improves a result ofJamison [2]. We also study convergence whenf is only measurable and give results related to the “zero-two” law. Finally two characterizations of uniform ergodicity for this class of operators are given. In particular a question ofLotz [4] is answered.

Uniform ergodic theorems for locally integrable semigroups and pseudoresolvents

We study uniform ergodicity (at ∞ \infty ) of a locally integrable operator semigroup T ( ⋅ ) T( \cdot ) of type w 0 {w_0} under a suitable condition which is weaker than the usual one w 0 ⩽ 0 {w_0} \leqslant 0 . We also give a precise characterization of the uniform Cesàro-ergodicity for semigroups of class ( 0 , A ) (0,A) . To prove the part of Abel-ergodicity we first prove a general uniform ergodic theorem for pseudo-resolvents.

Exponential Moments for Hitting Times of Uniformly Ergodic Markov Processes

Let $\mu$ be an invariant measure for a Markov process which is assumed $\mu$-uniformly ergodic in the following sense: the corresponding semigroup of operators on $L^2(d\mu)$, say $\{P_t; t \geq 0\}$, is such that the time average $(1/T) \int^T_0 P_t dt$ converges to a rank one projection in the uniform norm of operators. We prove that hitting times of sets having non zero $\mu$-measure possess moment generating functions.

Strong ergodicity for continuous-time, non-homogeneous Markov chains

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

Strong ergodicity for continuous-time, non-homogeneous Markov chains

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

On the Uniform Ergodic Theorem of Lin

Lin has given necessary and sufficient conditions for convergence in the uniform operator topology of ${A_n} = (1 + T + \cdots + {T^{n - 1}})/n$, $T$ being a Banach space operator satisfying $\left \| {{T^n}} \right \|/n \to 0$. We prove a generalization in which the Cesà ro means are replaced by any bounded sequence in the affine hull converging uniformly to invariance. In the case where $T:{C_0}(X) \to {C_0}(X)$ is a transient Feller operator for noncompact locally compact Hausdorff space $X$, we show that $\{ {A_n}\}$ converges strongly but never uniformly.

Über güteeigensehaften von tests bei homogenen markovschen prozessen

The paper deals with statistical tests of hypotheses about the real parameter of a homogeneous time-discrete Markov process. The power function of an asymptotically uniformly most powerful unbiased sequence of tests is approximately calculated and the convergence order of the remainder term is given. The essential assumption used is the uniform ergodicity of the Markov process.

Quasi-compactness and uniform ergodicity of positive operators

The following converse to the Yosida-Kakutani theorem is proved: IfT is a positive operator on a Banach lattice with ‖Tn‖/n → 0, thenT is quasi-compact if (and only if) the averages of its iterates converge uniformly to a finite-dimensional projection.

Ergodicity Using Mean Visit Times

The finiteness of the mean visit time to state j is used in the characterization of uniform strong ergodicity for non-stationary Markov chains and weak ergodicity for stationary Markov chains. This clarifies a key difference between ergodicity and weak ergodicity for infinite stochastic matrices.

Uniform ergodicity and strong mixing

as n---,oe. It should be noted that what we call strong mixing in this paper is a more stringent condition than what is ordinarily referred to as strong mixing, namely that P(A c~ zkB) ~P(A) P(B) as k ~ o o for any two measurable sets A, B. It is clear that strong mixing implies uniform ergodicity. In the case of stationary Markov sequences these conditions can be stated in another convenient form. If {X,} is a Markov sequence let d be the Borel field of measurable sets on the state space of the process. Let P ( , . ) be the one-step transition probability function of the Markov sequence and set

Conditional Probability Operators

The study of laws of random variables is facilitated by various methods of representing these laws. The distribution function and characteristic function have played an important role, and the functional representation of the law of a random variable $X$ as a mapping $T$ of the bounded Borel functions into the real line given by $Tg = Eg(X)$ connects many problems in probability theory with functional analysis. Similarly, in the study of conditional probability laws various representations of these conditional laws are desirable. In particular, in the study of Markov processes an operator representation has proven most useful. In this paper we develop the representation of the conditional law of a random variable in a way analogous to the functional representation mentioned above. Thus, if $X$ is a random variable on a probability space $(\Omega, \mathcal{A}, P)$ and $\mathcal{A}^\ast$ is a sub-$\sigma$-field of $\mathcal{A}$, then we introduce the $\mathcal{A}^\ast$ conditional operator $T$ of $X$ mapping the bounded Borel functions into $L_\infty(\Omega, \mathcal{A}^\ast, P)$ and given by $T_g = E^{\mathcal{A}\ast} g(X)$. The first four sections of the paper develop a rudimentary theory of such operators. The use of these operators in probability theory leads one to consider various operator topologies, etc., weaker than those usually studied in functional analysis. Most of the material presented in this development is quite elementary, and many properties of these operators having potential interest have not even been mentioned. Nevertheless, it is hoped that this exposition may suggest further use of the operator representation of conditional distributions in probability theory, and that the properties of these operators relevant to probability theory will be investigated more systematically. In the final section conditional probability operators are applied to a mixing problem. A stationary process is said to have central structure if, under conditions similar to those of the central limit problem for independent random variables, conditional limit laws given the invariant $\sigma$-field are infinitely divisible. It is shown that central structure is closely related to a type of uniform ergodicity.

Means for the bounded functions and ergodicity of the bounded representations of semi-groups

1. A mean on a semi-group 2 is a positive linear functional of norm one on the space m(z) of bounded, real-valued functions on 1. A bounded semigroup S of linear operators from a Banach space B to itself is called ergodic if there exists a system cd of averages A such that for every S in s limA (AS-A)=limA (SA -A) =0; we have three strengths of ergodicity of S according as uniform, strong, or weak convergence is used in the operator algebra. The first part of this paper deals with the relationship between existence of invariant means and ergodicity of bounded representations. In Theorem 2 it is shown that weak ergodicity of every bounded representation of I is equivalent to weak ergodicity of the right and left representations of f by right and left translations on m(2), and equivalent to the existence of a mean on m(z) invariant under right and left translations. These conditions, in turn, are equivalent to existence of a directed system of finite means on mn(z) converging weakly to two-sided invariance under all right and left translations of m(z). Uniform ergodicity is similarly related to existence of finite means converging in the norm of m(z2)* to two-sided invariance (Theorem 4). The second part of this paper gives some sufficient conditions for existence of invariant means or of finite means converging in norm to invariance. For the former, the Markoff method of proof by fixed-point arguments is applied (?5) to semi-groups and groups, and to semi-groups which are the union of expanding directed systems of sub-semi-groups with means. (For example, a group G such that every finite subset generates a finite subgroup has an invariant mean.) It is also proved by a direct construction (Theorem 6) that if G is a normal subgroup of a group H and if G and H/G have two-sided invariant means, so has H. In ?6 a parallel result is proved for finite means converging in norm to invariance. These results greatly increase the family of groups known to have invariant means. Solvable groups formed the only such class previously known; ?6 now shows that a solvable group satisfies a stronger property; it has a system of finite means converging in norm to invariance.